Success-driven distribution of public goods promotes cooperation but preserves defection

Matjaž Perc

Phys. Rev. E 84, 037102 – Published 9 September 2011

Abstract

Established already in Biblical times, the Matthew effect refers to the fact that in societies the rich tend to get richer and the potent even more powerful. Here we investigate a game theoretical model describing the evolution of cooperation on structured populations where the distribution of public goods is driven by the reproductive success of individuals. Phase diagrams reveal that cooperation is promoted irrespective of the uncertainty by strategy adoptions and the type of interaction graph, yet the complete dominance of cooperators is elusive due to the spontaneous emergence of superpersistent defectors that owe their survival to extremely rare microscopic patterns. This indicates that success-driven mechanisms are crucial for effectively harvesting benefits from collective actions but that they may also account for the observed persistence of maladaptive behavior.

Received 13 April 2011

DOI:https://doi.org/10.1103/PhysRevE.84.037102

Authors & Affiliations

Matjaž Perc

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia

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Vol. 84, Iss. 3 — September 2011

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Images

Figure 1
Critical $R$ at the $D \to C + D$ phase transition vs $Δ$ for three different values of $K$ . Success-driven distribution of public goods strongly decreases the multiplication factor needed for cooperation to survive, irrespective of $K$ . The inset depicts the probability of strategy adoption $w$ as a function of the payoff difference for the three values of $K$ [same color (gray scale) as in the main panel]. The square lattice with $G = 5$ was used as the interaction graph.Reuse & Permissions
Figure 2
Critical $R$ at the $D \to C + D$ phase transition vs $Δ$ for three different interaction graphs. The insets show the graphs schematically in the color (symbol type) corresponding to the results in the main panel, with the group size $G$ indicated for vertices encircled in red (gray). As in Fig. 1 the promotion of cooperation is clearly inferable, irrespective of the properties of the interaction graph. Here $K = 0.1$ was used in all three cases. The dashed orange line shows the result as predicted by the mean-field approximation (see main text for details).Reuse & Permissions
Figure 3
Strategy regions in the $Δ - R$ parameter space for the square lattice at $K = 0.1$ . While $R$ needed for cooperation to survive decreases steadily with increasing $Δ$ (solid red line), the opposite holds for the extinction of defectors. The transition to the pure $C$ region (dotted green line) is preceded by the emergence of a peculiar $C^{+ D}$ region (dashed blue line), where a minute fraction of defectors is able to survive in the presence of dominating cooperators. The inset shows two characteristic time courses for just above [dashed green line (top arrow at $R = 1.5$ )] and below [solid blue line (bottom arrow at $R = 1.47$ )] the $C^{+ D} \to C$ transition line for $Δ = 0.001$ . Within the $C^{+ D}$ region defectors utilize their one-in-a-million survival chance to evade extinction. Although in the inset only the first 4096 MCS are depicted, we have verified that at $R = 1.47,$ $ρ_{D} > 0$ for up to $10^{7}$ MCS at $L = 1600$ system size. The dashed black line in the inset is proportional to MCS $^{- 1}$ .Reuse & Permissions
Figure 4
The uppermost panel shows time courses of $M_{x}$ for the “superpersistent” defector (solid black line), its four nearest neighbors (dashed red line), its four next-nearest neighbors (dash-dotted green line), and the four next-next-nearest neighbors (dotted blue line). The three middle panels depict gray-scale-coded (with white maximal and black minimal) values of $M_{x}$ around the “superpersistent” defector at times corresponding to the vertical dashed lines in the uppermost panel. The colored stars serve to identify the neighbors corresponding to the colors (shades of gray) used in the uppermost panel. The bottommost row depicts possible configurations of defectors (gray) around the “superpersistent” defector (black). Results were obtained on a square lattice of linear size $L = 1600$ at $K = 0.1$ , $R = 1.47$ , and $Δ = 0.001$ .Reuse & Permissions

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